Yesterday we learned about expontentials and logarithms.We learned how to write an expontential fuction. Expontential function- is an equation of the form Y=ab* RESTRICTIONS-a cant equal 0,b has to be greater than 0,and b cant be equal to 1. We also learned how to write an expontential given two ordered pairs. The steps for solving are: 1.plug-in ordered pairs 2. solve for a 3.set equal and solve for b. 4. substitute b into one of the original equations to find a.
Shortcuts for graphing y-ab* 1. a is always the y-intercept. 2. b is known as the "stretch factor" Example:y=1(2) a b a=y-intercept,b=gtrowth - X-axis is asymptote, line is y=0. -Growth=graph up,Decay=graph down Example: 1.Y=3x, growth 2.Y=1.5x,growth 3.Y=.33x,decay 4.Y=153.7x,growth 5.Y=(1/5)x,decay *reminder:b greater than 1= growth b less than 1=decay Summary: if b=2, "parent graph" if b is greater than 0 and less than 1 is a proper fraction. Y-intercept is (0,a). Example:y=2x =(0,1) lastly,a vital reminder-ANY # RAISED TO THE ZERO POWER IS ONE!!!
In order to learn about exponential equations and how to sketch graphs of exponential functions, you first need to understand how to write and graph an exponential function and use that function to model exponential growth and decay. To begin, start by knowing that an exponential function is in the form Y=ab to the X power. The restrictions include that your "a" value cannot be equal to zero, and your b value has to be greater than zero but not equal to one. It is good to know shortcuts for grahing the functions. Begin by knowing that your "a" value is always the y-intercept of the graph and your "b" value is known as the "stretch" or growth factor. Also, always notice that the larger the "b" value gets, the more it stretches toward the y-axis and if "b" is a number between 0 and 1 the graph is decay. How do you write an exponential function given two ordered pairs? Basically it is a step by step process. Begin by using the given ordered pairs to replace x and y in each equation. For example, if you had the ordered pairs (2,2) and (3,4), you would substitute those values into your equation y=ab to the x power and solve for "a." Once you solve for "a" set the values equal to each other and cross multiply. After you cross multiply solve for your "b" factor. When solving for b is completed plug that factor back in to one of your equations for "a" to receive your "a" factor.Once all that is completed plug your values back into the equation Y=ab to the x power for your answer. If you remember all these steps for exponential functions and graphs your learning and understanding of this lesson should be very easy.
8.1A- In this lesson we learned how to write and graph exponential functions
An exponential function is (y-ab^x) In these there are a few restrictions: a can not = 0, b has to be greater than 0 and it cannot be 1.
There is a shortcut for graphing this equation: -a is always the y-intercept of the graph -b is known as the "strech" Factor or exponential growth -the larger b gets is the "strech" factor grows i.e.: y=(1)(2)^x - if b is less than 1 than the graph will experience exponential decay.
Below is an Example of a harder problem: (2,2);(3,4) plug in these numbers in the equation then the exponential equation y=ab^x 2=ab^2 and 4=ab^3; then solve for b 2/b^2=4/b^3; then solve for b b=2; then plug b back into one of the orginal "set equal" equations to get a. Then write the equation. y=1/2(2)^x
Something else we learned was an Asymptote- imagary line that acts as a boundry as what it gets close to but never reaches the x-axis.
I teach AP Calculus, Pre AP Precalculus, and Advanced Algebra/Trigonometry at Muscle Shoals High School. In my spare time, I enjoy spending time with my family and granddaughters. My husband and I like to travel and visited Germany and France last summer. I also enjoy reading, art and music.
Last spring, I read The Other Boleyn Girl by Philippa Gregory. I am currently reading the Constant Princess (by the same author).
I love "anything about England".
"SUGGESTED READING LIST" The Mathematical Traveler. It is a great book that combines history and interesting facts about famous mathematicians. See me if you want to "check it out".
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Yesterday we learned about expontentials and logarithms.We learned how to write an expontential fuction.
Expontential function- is an equation of the form Y=ab*
RESTRICTIONS-a cant equal 0,b has to be greater than 0,and b cant be equal to 1.
We also learned how to write an expontential given two ordered pairs. The steps for solving are:
1.plug-in ordered pairs
2. solve for a
3.set equal and solve for b.
4. substitute b into one of the original equations to find a.
Shortcuts for graphing y-ab*
1. a is always the y-intercept.
2. b is known as the "stretch factor"
Example:y=1(2)
a b
a=y-intercept,b=gtrowth
- X-axis is asymptote, line is y=0.
-Growth=graph up,Decay=graph down
Example:
1.Y=3x, growth
2.Y=1.5x,growth
3.Y=.33x,decay
4.Y=153.7x,growth
5.Y=(1/5)x,decay
*reminder:b greater than 1= growth
b less than 1=decay
Summary:
if b=2, "parent graph"
if b is greater than 0 and less than 1 is a proper fraction.
Y-intercept is (0,a).
Example:y=2x
=(0,1)
lastly,a vital reminder-ANY # RAISED TO THE ZERO POWER IS ONE!!!
In order to learn about exponential equations and how to sketch graphs of exponential functions, you first need to understand how to write and graph an exponential function and use that function to model exponential growth and decay.
To begin, start by knowing that an exponential function is in the form Y=ab to the X power. The restrictions include that your "a" value cannot be equal to zero, and your b value has to be greater than zero but not equal to one.
It is good to know shortcuts for grahing the functions. Begin by knowing that your "a" value is always the y-intercept of the graph and your "b" value is known as the "stretch" or growth factor. Also, always notice that the larger the "b" value gets, the more it stretches toward the y-axis and if "b" is a number between 0 and 1 the graph is decay.
How do you write an exponential function given two ordered pairs? Basically it is a step by step process. Begin by using the given ordered pairs to replace x and y in each equation. For example, if you had the ordered pairs (2,2) and (3,4), you would substitute those values into your equation y=ab to the x power and solve for "a."
Once you solve for "a" set the values equal to each other and cross multiply. After you cross multiply solve for your "b" factor. When solving for b is completed plug that factor back in to one of your equations for "a" to receive your "a" factor.Once all that is completed plug your values back into the equation Y=ab to the x power for your answer.
If you remember all these steps for exponential functions and graphs your learning and understanding of this lesson should be very easy.
8.1A-
In this lesson we learned how to write and graph exponential functions
An exponential function is (y-ab^x)
In these there are a few restrictions: a can not = 0, b has to be greater than 0 and it cannot be 1.
There is a shortcut for graphing this equation:
-a is always the y-intercept of the graph
-b is known as the "strech" Factor
or exponential growth
-the larger b gets is the "strech" factor grows
i.e.: y=(1)(2)^x
- if b is less than 1 than the graph will experience exponential decay.
Below is an Example of a harder problem:
(2,2);(3,4) plug in these numbers in the equation
then the exponential equation y=ab^x
2=ab^2 and 4=ab^3; then solve for b
2/b^2=4/b^3; then solve for b
b=2; then plug b back into one of the orginal "set equal" equations to get a. Then write the equation.
y=1/2(2)^x
Something else we learned was an Asymptote- imagary line that acts as a boundry as what it gets close to but never reaches the x-axis.
Lesson 8.1
Y=ab^x is an exponential function in an equation.
a can't equal 0,b > 0,b can't equal
A decay graph is when 0 < b < 1.
A growth graph is when b > 1.
Shortcuts for graphing y=ab^x
a is always the y-intercept of the graph
b is known as the "stretch factor"
Asymtote is x-axis, y=0
Y intercept is (0,1)
An example of a Exponential Function Y=ab^x
(-1,8/1/3)&(2,1.8) orgional problem
(-1,25/3)&(2,9/5) change to improper fractions
25/3=ab^-1 9/5=ab^2 sub. the x & y values
25/3=a b/b (the bs cancel) 9/5b^2=ab^2/b^2(b^2s cancel on the right side)
25b/3=a 9/5b^2=a
(5b^2)25b/3=9/5b^2(5b^2) (the 5b^2 cancel on the right side)
3/125(125b3/3)=9(3/125) (everything on the left side cancels except the b^3)
3square root b^3=3square root 27/125
= b=3/5 (simplify)
sub. your answers in the equation y=ab^x
THE ANSWER y=(9/5)*(3/5)^x
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